Question

Calculate the speed of sound at 288 K in hydrogen, helium, and nitrogen. Under what conditions will the speed of sound in hydrogen be equal to that in helium?

Answer 1

Explanation:

The sped of sound is given as follows.

            C = $\sqrt{\gamma RT}$

It is known that for hydrogen,

         R = 4124 J/kg K

         T = 288 k

       $\gamma$ = 1.41

Therefore, calculate the value of $C_{hydrogen}$ as follows.

         $C_{hydrogen} = \sqrt{\gamma RT}$

                     = $\sqrt{1.41 \times 4124 J/kg K \times 288}$

                     = 1294.1 m/s

For helium,

         R = 2077 J/kg K

         T = 288 k

       $\gamma$ = 1.66

Therefore, calculate the value of $C_{helium}$ as follows.

         $C_{helium} = \sqrt{\gamma RT}$

                     = $\sqrt{1.66 \times 2077 J/kg K \times 288}$

                     = 996.48 m/s

For nitrogen,

         R = 296.8 J/kg K

         T = 288 k

       $\gamma$ = 1.4

Therefore, calculate the value of $C_{hydrogen}$ as follows.

         $C_{hydrogen} = \sqrt{\gamma RT}$

                     = $\sqrt{1.4 \times 296.8 J/kg K \times 288}$

                     = 345.93 m/s

So, speed of sound in hydrogen is calculated as follows.

            = $\sqrt{1.41 \times 4124 \times T_{H}}$

            = $76.26 \sqrt{T_{H}}$

Speed of sound in helium is as follows.

            = $\sqrt{1.66 \times 2077 \times T_{He}}$

            = $58.72 \sqrt{T_{He}}$

For both the speeds to be equal,

       $76.26 \sqrt{T_{H}}$ = $58.72 \sqrt{T_{He}}$

        $\frac{T_{H}}{T_{He}}$ = 0.593

Therefore, we can conclude that the temperature of hydrogen is 0.593 times the temperature of helium.